1. 1-4 Extrema and Average Rates of Change

2. Determine whether the function is continuous at x = 4.
A. yes
B. no
5–Minute Check 2

3. Determine whether the function is continuous at x = 2.
A. yes
B. no
5–Minute Check 3

4. Describe the end behavior of f (x) = –6x 4 + 3x 3 – 17x 2 – 5x + 12.
5–Minute Check 4

5. Determine between which consecutive integers the real zeros of f (x) = x 3 + x 2 – 2x + 5 are located on the interval [–4, 4].
A. –2 < x < –1
B. –3 < x < –2
C. 0 < x < 1
D. –4 < x < –3
5–Minute Check 5

6. You found function values. (Lesson 1-1)
Determine intervals on which functions are increasing, constant, or decreasing, and determine maxima and minima of functions.
Determine the average rate of change of a function.
Then/Now

7. Key Concept 1

8. As x increases, f(x) increases
Key Concept 1

9. As x increases, f(x) decreases
Key Concept 1

10. As x increases, f(x) stays the same
Key Concept 1

11. Analyze Increasing and Decreasing Behavior
A. Use the graph of the function f (x) = x 2 – 4 to estimate intervals to the nearest 0.5 unit on which the function is increasing, decreasing, or constant. Support the answer numerically.
Example 1

12. Support Numerically – (for demonstration)
Analyze Increasing and Decreasing Behavior
Analyze Graphically
From the graph, we can estimate that f is decreasing on and increasing on
Support Numerically – (for demonstration)
Create a table using x-values in each interval.
The table shows that as x increases from negative values to 0, f (x) decreases; as x increases from 0 to positive values, f (x) increases. This supports the conjecture.
Example 1

13. Answer: f (x) is decreasing on and increasing on .
Analyze Increasing and Decreasing Behavior
Answer: f (x) is decreasing on and increasing on
Example 1

14. Analyze Increasing and Decreasing Behavior
B. Use the graph of the function f (x) = –x 3 + x to estimate intervals to the nearest 0.5 unit on which the function is increasing, decreasing, or constant. Support the answer numerically.
Example 1

15. Create a table using x-values in each interval.
Analyze Increasing and Decreasing Behavior
Analyze Graphically
From the graph, we can estimate that f is decreasing on , increasing on , and decreasing on
Support Numerically
Create a table using x-values in each interval.
Example 1

16. Analyze Increasing and Decreasing Behavior
0.5 1 2
2.5 3 –6
–13.125
–24
Example 1

17. Answer: f (x) is decreasing on and and increasing on
Analyze Increasing and Decreasing Behavior
The table shows that as x increases to , f (x) decreases; as x increases from , f (x) increases; as x increases from , f (x) decreases. This supports the conjecture.
Answer: f (x) is decreasing on and
and increasing on
Example 1

18. A. f (x) is increasing on (–∞, –1) and (–1, ∞).
Use the graph of the function f (x) = 2x 2 + 3x – 1 to estimate intervals to the nearest 0.5 unit on which the function is increasing, decreasing, or constant. Support the answer numerically.
A. f (x) is increasing on (–∞, –1) and (–1, ∞).
B. f (x) is increasing on (–∞, –1) and decreasing on (–1, ∞).
C. f (x) is decreasing on (–∞, –1) and increasing on (–1, ∞).
D. f (x) is decreasing on (–∞, –1) and decreasing on (–1, ∞).
Example 1

19. QUESTIONS?

20. Key Concept 2

21. Key Concept 2

22. Estimate and Identify Extrema of a Function
Estimate and classify the extrema to the nearest 0.5 unit for the graph of f (x).
Example 2

23. Estimate and Identify Extrema of a Function
Analyze Graphically
It appears that f (x) has a relative minimum at x = –1 and a relative maximum at x = 2. It also appears that so we conjecture that this function has no absolute extrema.
Answer: To the nearest 0.5 unit, there is a relative minimum at x = –1 and a relative maximum at x = 2. There are no absolute extrema
Example 2

24. Use a Graphing Calculator to Approximate Extrema
GRAPHING CALCULATOR Approximate to the nearest hundredth the relative or absolute extrema of f (x) = x 4 – 5x 2 – 2x + 4. State the x-value(s) where they occur.
f (x) = x 4 – 5x 2 – 2x + 4
Graph the function and adjust the window as needed so that all of the graph’s behavior is visible.
Example 3

25. Use a Graphing Calculator to Approximate Extrema
Answer: relative minimum: (–1.47, 0.80); relative maximum: (–0.20, 4.20); absolute minimum: (1.67, –5.51)
Example 3

26. Day 2
Average Rate of Change

27. Key Concept3

28. Substitute –3 for x1 and –1 for x2.
Find Average Rates of Change
A. Find the average rate of change of f (x) = –2x 2 + 4x + 6 on the interval [–3, –1].
Use the Slope Formula to find the average rate of change of f on the interval [–3, –1].
Substitute –3 for x1 and –1 for x2.
Evaluate f(–1) and f(–3).
Example 5

29. Homework QUIZ TOMORROW!!!
Pg 40: 9-13, 21, 24, 28, 40, 42, 47, 54-56, 60-63
QUIZ TOMORROW!!!